Journal of Statistical Mechanics Theory and Experiment

Quasi-stationary-state temperature for interacting Heisenberg rotators: a finite-size scaling analysis

Antonio Rodríguez, Fernando Dantas Nobre, Constantino Tsallis

Published Feb 1, 2026• 0 citations

Abstract

Abstract We study, by means of first-principle molecular-dynamics, the temperature T QSS of the quasi-stationary state (QSS) of a d -dimensional classical inertial α -Heisenberg ferromagnetic model, consisting of a set of rotators with interactions decaying with distance as r − α ( α ⩾ 0 ) . We find, through detailed finite-size calculations, the scaling relation T QSS − T ∞ ∝ N − β , where T ∞ ≡ U − 1 / 2 , with energy per particle U < U c ( U c = 5 / 6 ), and total number of particles N , which holds for the very long-range interactions regime 0 ⩽ α / d ⩽ 1 , in which the QSS exists. We show that the scaling exponent β does not depend on U but only in the quotient α / d , being β ( α / d ) ≃ 1 for 0 ⩽ α / d ⩽ 0.5 , and decaying linearly for α / d > 0.5 , so <

DOI: 10.1088/1742-5468/ae3a6d

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Journal of Statistical Mechanics Theory and Experiment

Quasi-stationary-state temperature for interacting Heisenberg rotators: a finite-size scaling analysis

Antonio Rodríguez, Fernando Dantas Nobre, Constantino Tsallis

Published Feb 1, 2026• 0 citations

Publisher

Abstract

DOI: 10.1088/1742-5468/ae3a6d

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The duration of the quasistationary states (QSSs) emerging in the

d

-dimensional classical inertial $\ensuremath{\alpha}$-Heisenberg model, i.e.,

N

three-dimensional rotators whose interactions decay with distance

{r}_{ij}

as $1/{r}_{ij}^{\ensuremath{\alpha}}$ ($\ensuremath{\alpha}\ensuremath{\ge}0$), is studied through first-principle molecular dynamics. These QSSs appear for the very-long-range interaction regime ($0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$), for an average energy per rotator

U<{U}_{c}

(

{U}_{c}=5/6

), and they do not exist for

U>{U}_{c}

. They are characterized by a kinetic temperature

{T}_{\text{QSS}}

, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature

{T}_{\text{BG}}>{T}_{\text{QSS}}

. We investigate here the behavior of their duration

{t}_{\text{QSS}}

when

U

approaches

{U}_{c}

from below, for large values of

N

. The QSS gradually disappears as $U\ensuremath{\rightarrow}{U}_{c}$, while its duration undergoes a critical phenomenon, namely ${t}_{\text{QSS}}\ensuremath{\propto}{({U}_{c}\ensuremath{-}U)}^{\ensuremath{-}\ensuremath{\xi}}$. Universality is found for the critical exponent $\ensuremath{\xi}=1\text{.}67\ifmmode\pm\else\textpm\fi{}0\text{.}02$ throughout the very-long-range interaction regime.

We numerically study, from first principles, the temperature

{T}_{\mathrm{QSS}}

and duration

{t}_{\mathrm{QSS}}

of the longstanding initial quasi-stationary state of the isolated

d

-dimensional classical inertial $\ensuremath{\alpha}\text{\ensuremath{-}}XY$ ferromagnet with two-body interactions decaying as $1/{r}_{ij}^{\ensuremath{\alpha}}$ $(\ensuremath{\alpha}\ensuremath{\ge}0)$. It is shown that this temperature

{T}_{\mathrm{QSS}}

(defined proportional to the kinetic energy per particle) depends, for the long-range regime $0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$, on $(\ensuremath{\alpha},d,U,N)$ with numerically negligible changes for dimensions

d=1,2,3

, with

U

the energy per particle and

N

the number of particles. We verify the finite-size scaling ${T}_{\mathrm{QSS}}\ensuremath{-}{T}_{\ensuremath{\infty}}\ensuremath{\propto}1/{N}^{\ensuremath{\beta}}$ where ${T}_{\ensuremath{\infty}}\ensuremath{\equiv}2U\ensuremath{-}1$ for $U\ensuremath{\lesssim}{U}_{c}$, and $\ensuremath{\beta}$ appears to depend sensibly only on $\ensuremath{\alpha}/d$. Our numerical results indicate that neither the scaling with

N

{T}_{\mathrm{QSS}}

nor that of

{t}_{\mathrm{QSS}}

depend on

U

Quantum Heisenberg ferromagnets with long-range interactions in low dimensions are inves- tigated by means of the Green's frunction method. The model Hamiltonian is given by scrH=-(

{\mathit{J}}_{0}

/2)${\mathcal{J}}_{\mathit{i}\mathrm{\ensuremath{\ne}}\mathit{j}}$${\mathit{r}}_{\mathit{i}\mathit{j}}^{\mathrm{\ensuremath{-}}\mathit{p}}$

{\mathbf{S}}_{\mathit{i}}

\ensuremath{\cdot}

{\mathbf{S}}_{\mathit{j}}

-H

{\mathcal{J}}_{\mathit{j}}

{\mathit{S}}_{\mathit{j}}^{\mathit{z}}

. It is shown that there exists a finite-temperature phase transition in the region dp2d for the d-dimensional case and that no transitions at any finite temperature exist for p\ensuremath{\ge}2d; the critical temperature is also estimated. We study the magnetic properties of this model. We calculate the critical exponents' dependence on p; these exponents also satisfy a scaling relation. Some of the results were also found using the modified spin-wave theory and are in remarkable agreement with each other.